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Neuropsychologia 44 (2006) 2972–2986

Acalculia from a right hemisphere lesionDealing with “where” in multiplication procedures

Alessia Grana a,b,∗, Riccardo Hofer a, Carlo Semenza a

a Dipartimento di Psicologia, Universita degli Studi di Trieste, via S. Anastasio 12, 34124 Trieste, Italyb Istituto di Medicina Fisica e Riabilitazione, via Gervasutta 48, 33100 Udine, Italy

Received 26 September 2005; received in revised form 12 June 2006; accepted 18 June 2006Available online 17 August 2006

bstract

The present study describes in detail, for the first time, a case of failure with multiplication procedures in a right hemisphere damaged patientPN). A careful, step-by-step, error analysis made possible to show that an important portion of PN’s errors could be better explained as spatial inature and specifically related to the demands of a multi-digit multiplication. These errors can be distinguished from other types of errors, includinghose, expected after a right hemisphere lesion, determined by a generic inability to deal with spatial material, or from other deficits, like neglect,ffecting cognitive capacities across the board.

The best explanation for PN’s problems is that he might have difficulties in relying on a visuo-spatial store containing a layout representationpecific to multiplication. As a consequence, while knowing what, when and how to carry out the various steps, PN does not know where. What heay thus lack is a spatial schema of multiplication. Such schema is thought to help normal calculators in overcoming working memory demands

f complex calculation by representing the information of where exactly each sub-step should be placed.2006 Elsevier Ltd. All rights reserved.

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eywords: Spatial layout for multiplication; Multi-digit written calculation; Pa

. Introduction

The present study describes in detail, for the first time, a casef failure with multiplication procedures in a right hemisphereamaged patient. In this context, problems will be addressedrising with the current distinction between primary and sec-ndary acalculia and with the definition of spatial acalculia. Anffort will be made at starting to identify which and to whatxtent sub-components of the calculation process may need theupport of spatial cognition.

.1. The nature of deficits in arithmetical procedures andheir localisation

The pattern of impairment described in single case studiesf deficit in arithmetical procedures used in multi-digit calcu-ation varies from case to case. In a first effort, McCloskey,aramazza, and Basili (1985) had provided a few examples of a

∗ Corresponding author. Tel.: +39 040 5582725; fax: +39 040 4528022.E-mail address: [email protected] (A. Grana).

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028-3932/$ – see front matter © 2006 Elsevier Ltd. All rights reserved.oi:10.1016/j.neuropsychologia.2006.06.027

roducts in multiplication; Factor selection in multiplication; Spatial acalculia

ariety of errors with procedures, such as problems in carryingnd borrowing with misordering of steps, misalignment, fail-re of organising intermediate products, failure in integratingntermediate products, etc., in four patients whose full symp-omathology, however, was not reported.

More detailed case studies appear in the literature of the pastecade. Lucchelli and De Renzi (1993) described what wereostly errors on carrying procedures in a patient who was rel-

tively more proficient with arithmetical fact retrieval. In thisase a vascular lesion involved areas 10, 11, 16, 14, 23, 6 and 4nd anterior half of corpus callosum.

A similar dissociation was reported by Temple (1991) in aase of developmental dyscalculia. In patient MT, describedy Girelli and Delazer (1996), who also had problems withetrieving single-digit multiplication (which prevented studyingrocedures in multi-digit multiplication), a special kind of pro-edural error (“bug”) was identified in subtraction. This erroronsisted of the systematic application of a disturbed under-
ying algorithm: MT subtracted the smaller number from thearger one irrespective of the position. In this case the lesionas provoked by a left fronto-temporo-parietal cerebro-vascular
ccident.

mailto:[email protected]

dx.doi.org/10.1016/j.neuropsychologia.2006.06.027

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A. Grana et al. / Neuropsy

Recently Sandrini, Miozzo, Cotelli, and Cappa (2003) repli-ated this finding in a very severe fluent aphasic patient whouffered of a haemorrhage in the left temporo-parietal lobe.emenza, Miceli, and Girelli (1997) described patient MM,ffected by hydrocephalus (with more severe effects in the rightpper parietal and right dorso-frontal areas), who was insteaderfect with arithmetical facts but made unsystematic and pro-ressively more frequent procedural errors in multi-digit mul-iplication. MM, who always started in the correct way, hademarkable difficulties in ending the operation, and showed lit-le awareness about the precision of his performance. For theseeasons Semenza et al. proposed that MM was affected by amonitoring deficit”, that is an inability to monitor the sequencef operations that calculation procedures specify rather thaneflecting faulty knowledge of the algorithm to be implemented.n contrast, cases like Girelli and Delazer’s (1996), mentionedbove, could be characterised as having a memory deficit forprocedural algorithm. Grana, Girelli, and Semenza (2001)

eported another case (patient FS) who (following a traumaticrain injury damaging the left temporo-parietal areas and a smallrea in the fronto-parietal right hemisphere), again in presencef intact arithmetical facts knowledge, showed a selective pro-edural deficit in multi-digit operations. FS never decomposedntermediate products, thus leading to highly implausible resultshat he judged instead as plausible and correct. Finally, McNeilnd Burgess (2002) recently reported a further case of pro-edural deficit in a probable Alzheimer’s dementia. This caseresented similarities with MM, including preservation of arith-etical facts. None of the two cases however seem interpretable,

s suggested by McNeil and Burgess, only as a task-switchingroblem. At least in MM’s case most errors were found whenwitching was not required. Furthermore, task switching differ-ntiates multiplication from addition only in the final step theperation, i.e., when, after calculating partial products, theseust be added: if switching tasks were the problem, one would

xpect more errors at this point, which, at least in MM’s case,id not happen.

This brief perusal reveals that described cases derive fromither left sided or bilateral/diffused lesions. No single caseescription of procedure deficit is reported after damage con-ned to the right hemisphere.

.2. Calculation in the right hemisphere

Acalculia following right hemisphere lesions has been con-istently found in group studies, though in various proportions,robably due to the different sensitivity of different calculationests (e.g., Ardila & Rosselli, 1994; Basso, Burgio, & Caporali,000; Dahmen, Hartje, Bussing, & Sturm, 1982; Grafman,assafiume, Faglioni, & Boller, 1982; Hecaen, 1962; Hecaen &ngelergues, 1961). Hecaen, Angelergues, and Houiller (1961)

lassified 183 acalculic patients with retrorolandic lesionsccording to the clinical signs of: (a) difficulties in the reading
nd writing numerals: alexia or agraphia for number (34.5% ofhe cases); (b) difficulties in the execution of arithmetical oper-tions: anarithmetia (39.3% of the cases); (c) difficulties in thepatial arrangement of numbers: spatial acalculia. This latter
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ariety was assumed to be present in 26.2% and was essentiallyonsidered as secondary to visuo-spatial troubles (see below).he calculation disorder of spatial type was present in 23.5% ofatients with right retrorolandic lesions, in 17% of patients withilateral lesions and only in 2% of patients with left retrorolandicesions.

Grafman et al. (1982) also found that right hemisphere dam-ged patients were impaired in written multi-digit calculation,ut the type of errors were not analysed. Dahmen et al. (1982)ompared the performance of Broca’s aphasics, Wernicke’sphasics and right brain damaged subjects in a variety of dif-erent tasks, tapping what they termed verbal and visuo-spatialomponents of arithmetic tasks, not including multi-digit calcu-ation. Their main conclusion was that, since both Wernicke’sphasics and right hemisphere patients showed spatial disorders,patial disorders in Wernicke’s aphasia could not be secondaryo language impairment. Ardila and Rosselli (1994) tested 21atients with right hemisphere damage (6 pre-Rolandic and5 retro-Rolandic) with a “special calculation battery”. Theirndings suggest that acalculia appeared particularly in writtenalculation and would be better preserved in mental calculation.n reading and writing numbers they found spatial alexia andgraphia, leading to particular errors (feature and digit addition,nability to use the spaces to join and separate numbers, difficultyn conserving the written line in a horizontal position, increasedeft margins and unsteadiness in maintaining left margins, disre-pect of spaces and spatial disorganisation of written material).s far as the calculation system was concerned, they found lossf calculation automatisms and reasoning errors (impossibleesults were not rejected). It can be doubted that these errorsre specific to right brain damage. Ardila and Rosselli (1994)nvestigation was too limited to draw less than generic conclu-ions (e.g., written calculation was evaluated on only four items),ut can give at least some indications about what kind of errorsay appear in right hemisphere damaged subjects. Basso et al.

2000) analysed the deficit in number processing in 50 left and6 right brain damaged subjects. Patients were classified as acal-ulic if they failed in more than three tasks on the EC301 Batteryor Calculation and Number Processing. Following these crite-ia, and excluding patients whose symptoms could be affectedy unilateral neglect, Basso et al. diagnosed as acalculic only 3ut of 26 right brain damaged patients. However, the exact dataf these three patients on written calculation were not reported.ingle cases of acalculia in right brain damaged patients are

ndeed rare and rather anecdotal (e.g., Ardila & Rosselli, 2002;eleux, Kaiser, & Lebrun, 1979).

Bilateral activation, at least of the parietal lobes, is theost common finding in neuroimaging studies on calculation

Chochon, Cohen, van de Moortele, & Dehaene, 1999; Cohen,ehaene, Chochon, Lehericy, & Naccache, 2000; Dehaene,iazza, Pinel, & Cohen, 2003; Dehaene, Spelke, Pinel, Stanescu,Tsivkin, 1999; Eger, Sterzer, Russ, Giraud, & Kleinschmidt,

003; Hubbard, Piazza, Pinel, & Dehaene, 2005; Menon, Rivera,
hite, Glover, & Reiss, 2000; Naccache & Dehaene, 2001;
iazza, Mechelli, Butterworth, & Price, 2002; Pinel, Dehaene,iviere, & LeBihan, 2001; Rickard et al., 2000; Roland &riberg, 1985; Thioux, Pesenti, De Volder, & Seron 2002; Zago

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974 A. Grana et al. / Neuropsy

t al., 2001). Evidence of relatively more intense activation onhe right seems to be possibly related to proficiency (Zago et al.,001). More specifically, however, and of relevance to the inter-sts of the present study, no investigation so far has been maden complex written multiplication. It may be nonetheless usefulo mention in some detail the main results of an accurate reviewnd meta-analysis of recent imaging and patient studies on cal-ulation recently provided by Dehaene et al. (2003); see alsoehaene, Molko, Cohen, & Wilson (2004). The authors’ con-

lusion is that the intraparietal, angular and posterior parietalegions play distinct functional roles in arithmetic. In particu-ar, the horizontal segment of the bilateral intraparietal sulcus isystematically activated whenever numbers are processed, inde-endently of the notation used for numbers, and the activationncreases when the task requires the processing of quantities.his area is activated in calculation (Burbaud et al., 1999;hochon et al., 1999; Pesenti, Thioux, Seron, & De Volder,000; Rickard et al., 2000), more in approximate than in exactalculation (Dehaene et al., 1999) and more in subtraction thann multiplication (Chochon et al., 1999; Lee, 2000). The otherwo circuits involved in number processing, the left angular ver-al system and the posterior superior parietal attention system,ely on representations and processes that are not specific to theumber domain. Particularly, the left angular gyrus supports ver-al aspects of numerical processing (Simon, Mangin, Cohen, Leihan, & Dehaene, 2002; Stanescu-Cosson et al., 2000). It medi-tes the retrieval of facts stored in verbal memory, but not otherumerical tasks, like subtraction, number comparison or com-lex calculation, related to processing of quantity (Chochon etl., 1999; Delazer, Domahs et al., 2003; Duffau et al., 2002; Lee,000). The posterior superior parietal lobule, instead, supportsisuo-spatial processes, attention and spatial working mem-ry (Corbetta, Kincade, Ollinger, McAvoy, & Shulman, 2000;ulham & Kanwisher, 2001; Pesenti et al., 2000; Pinel et al.,001; Simon et al., 2002) related to numerical processing. It isctivated in number comparison (Pesenti et al., 2000; Pinel et al.,001), approximation (Dehaene et al., 1999) or counting (Piazzat al., 2002; Piazza, Giacomini, Le Bihan, & Dehaene, 2003).t also appears to increase in activation when subjects carry outwo operations instead of one (Menon et al., 2000). In conclu-ion, the role of different structures within the neural networkupporting complex mental calculation has been widely inves-igated but little is still known about the neural implementationf calculation when further increment in complexity requiresesorting to the written modality. However, in order to collecthis information in a sensible way, and before looking for precisenatomical locations, it is important to preliminarily disentan-le conceptually each component of calculation at this moreomplex level. To this aim the study of dissociations offeredy neuropsychological cases may indeed still be a fundamentalool.

.3. The contribution of spatial cognition to calculation: an
ld neuropsychological problem
The largest part of neuropsychological research on num-er processing and calculation concerns the verbal compo-

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gia 44 (2006) 2972–2986

ent. A strong relationship between language and numbers haseen found in numerous studies (e.g., Seron, 2001; Spelke

Tsivkin, 2001). However, evidence for the contrary is alson record. Hermelin and O’Connor (1990) found astoundingumerical abilities in a languageless autistic young man andossor, Warrington, and Cipolotti (1995) found a severely apha-

ic patient with left temporal lobe atrophy still being able toalculate (see Giaquinto, 2001). In contrast, the spatial compo-ent has been defined rather intuitively, and remains vaguely andmplicitly acknowledged (see Caramazza & McCloskey, 1987;

iceli & Capasso, 1999), being, paradoxically, relatively moreealt with, as clearly shown in an excellent review by Hartje1987), in the old neuropsychological literature than in moreecent investigations. Indeed, some of the historical papers (e.g.,eritz, 1918; e.g., Ehrenwald, 1931; Kleist, 1934; Krapf, 1937;eonhard, 1938; Singer & Low, 1933), based on single casesnd phenomenological observation, found a close relationshipetween space and arithmetical and numerical tasks.

The notion of “spatial acalculia” (SA) is nowadays generallysed to refer to calculation deficits that are intuitively consid-red to be secondary to spatial problems. In their seminal paper,ecaen et al. (1961) applied the label of “acalculia of spatial

ype” to a group of patients who were “unable to respect theosition and the order of digits in relation to one another”, andxtended the notion to include troubles to calculation derivingrom spatial neglect and from “spatial inversions of operationalechanisms”. The label spatial acalculia is thus given to patients

ailing in the visuo-spatial components of arithmetic tasks: aist of these failures appears in Hartje’s review, that also listshe types of spatial disorders that would affect calculation pro-essing. However, this list of putative errors lacks a link tompirical observations and does not clearly distinguish whichrrors should be considered secondary to visuo-spatial disorderse.g., visuo-spatial neglect) and which errors could primarilytem from a disorder of operation specific visuo-spatial mem-ry.

For the purpose of the present investigation only cases wherespatial disorder would affect in a visible way arithmetical

rocedures must be considered. In fact, in all previous stud-es on failure on arithmetical procedures, the errors in writtenperations were never classified according to the different oper-tion steps (i.e., in the case of multiplication, factor selection,rithmetical fact retrieval, carry, calculation of partial products,ddition of partial products). It thus remains unclear preciselyn which steps right hemisphere acalculic subjects err the mostnd why, and, in particular, to what extent the found deficits areecondary to visuo-spatial functions.

Models introduced by cognitive neuropsychology are so farf little help in the enterprise of disentangling spatial from non-patial components in complex calculation and of identifyingheir neural localisation. Even the “triple code model” proposedy Dehaene and Cohen (1995), which is the only one that tries toie specific cognitive processes to anatomical locations, does not
eally fill this gap of knowledge. In fact it only implies the basicomponents of number processing: none of these representationsan explain how humans are able to solve multi-digit complexultiplications, how much do they rely on spatial cognition and

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here is the algorithm for the different operations (addition,ubtraction, division and multiplication) stored in the brain.

One of the few relevant mentions of a specific spatial com-onent in multi-digit calculation may be found in Dehaene andohen (1995, p. 102): “A visuo-spatial store may also be used

o maintain on-line the spatial layout and digits of an ongo-ng multi-digit calculation”. On the same line, Hartje (1987)ad earlier observed how, in complex calculation, more or lessutomatised spatial patterns of processing interact with logicaleasoning: beyond a certain level of difficulty only very sophis-icated subjects could possibly perform complex calculationithout resorting to these patterns. However, these reasonable

ssumptions are not accounted for in Dehaene and Cohen’s1995) actual model and they were not systematically inves-igated ever since. Furthermore, no theory has ever been offeredbout the processes involved in “maintaining the spatial layoutn-line” and the LTM representation of the spatial layout itself.

In summary, available literature on the sort of calculation dis-rder that may be considered typical of a right hemisphere lesions far from being satisfactory. As Miceli and Capasso (1999)rgued, “. . . from the metatheoretical and the methodologicaltandpoint the reported analyses are insufficiently detailed, theriteria on which they are based are intuitive and are not justi-ed by clear hypotheses on the structure of the cognitive systemnder analysis”. The very concept of spatial acalculia is indeedmbiguously used. It is understood, as reported above, that inost cases, on an intuitive basis, spatial acalculia is considered

econdary to other disturbances. No effort has ever been made indentifying exactly which components of multi-digit calculation

ay rely on spatial cognition and which pattern of performancehould be expected in the case of a specific disturbance of theseunctions. For instance, Hartje (1987) mentions the fact that “inriting down the intermediate products or the number values toe carried, shifting the numbers to the right will cause errors”.s it possible to find such shift, and its consequences, indepen-ently from spatial disorders determining every configurationo shift to the right? In other words: is it possible to distinguishhen the patient shifts to the right because he/she has lost the

apacity to explore space from the case where the informationegarding the positions of the steps is inaccessible to the patientecause of a permanent loss or because of a refractory (or accessifficulty) state? No answer to problems of this sort has ever beenttempted.

.4. A starting point

A possible starting point may consist in simply acknowledg-ng, like Dehaene and Cohen (1995) did without pursuing thisssue any further, that some basic spatial processing components

ust be inherent to multi-digit written calculation as it is usuallyerformed. As already reported, Dehaene and Cohen, in particu-ar, mention a “spatial layout” that should be maintained on-line.his component must involve a long term memory representa-

ion, specifying what needs to be worked upon and, crucially,here. This representation, indeed, must be maintained on-line

n order to perform the calculation. However, beyond a certainevel of complexity, the actual calculation may exceed work-

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ng memory capacities, and writing it down in its componentsccording to a conventional procedure, respecting, in particular,conventional spatial layout, may become necessary to perform

he operation. In order to minimise the effort, long term mem-ry must therefore contain not only the information of what isorked upon and how, but also of where the actual figures need

o be written. Mastering this information (mentioned in Hartje asautomatised spatial patterns of processing”) ultimately makesomplex calculation more automatic, less in need of continuousack-up and therefore more easily accessible to the calculatingndividual. Indeed, in principle, there is no need to appeal tospatially encoded, pre-organised schema to facilitate the pro-

edure. It is however hard to believe that the very process ofriting down the calculation according to pre-learned conven-

ional order and display may entirely dispense with the help ofpatially coded information.

The calculating individual may be, as a result of several causalactors, like brain damage, poor genetic equipment, insufficientraining and other unfavourable conditions, unable to lay downhe written operation. This may happen either because of annability to use spatial information properly (e.g., in the instancef disturbances like neglect) or, more specifically, because of aelatively selective disturbance to the stored memory for theroper layout or to its retrieval. These alternatives are, at leastn principle, distinguishable. Indeed, it may ultimately be veryifficult if possible at all to capture and disentangle preciselyll factors determining success in correctly carrying out a com-lex multi-digit multiplication. However, it does not seem a goodpproach to pretend, vis-a-vis these difficulties, that informationf spatial nature, stored in order to simplify complex multipli-ation procedures, does not exist.

An answer to the question of whether this analysis reflectssychological reality may come from the observation, carriedut with the methods of cognitive neuropsychology, of patientsffected by acalculia. The case of a patient who, as a result ofight hemisphere damage, has lost the ability to carry on calcu-ation procedures lends itself to such an investigation.

. Case report

PN was a 32-year-old right-handed businessman with 15ears of formal education. He was pre-morbidly very familiarith simple mathematics, in particular with multi-digit calcula-

ion, that he currently used in his business. In December 2002 heuffered from a haemorrhage as a consequence of the rupture ofright middle cerebral artery aneurysm, which was then surgi-

ally treated. A CT scan performed in February 2003 showed aast hemorrhagic intraparenchimal lesion in the right temporo-arietal region (see Fig. 1), involving the polar planum, Heschlyrus, the temporal planum, posterior insula, the supramarginalyrus and the angular gyrus, with dilatation of the right lateralentriculum; no lesion in the left hemisphere was detectable.e was left with a very mild left hemiparesis, left hemianopsia

nd mild cognitive deficits. The investigation on PN’s impair-ent with procedures in multi-digit calculation was carried out

n two different periods, first from February to mid-March 2003,nd, second, from the end of March to the end of April 2003.

2976 A. Grana et al. / Neuropsychologia 44 (2006) 2972–2986

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efore each of these two periods PN underwent a general assess-ent of cognitive functions, including a routine evaluation of hisathematical skills. He gave his informed consent to the present

nvestigation.

. Methods

PN’s performance on multi-digit multiplications was compared with that ofhree age- and education-matched controls in order to better match the qualityf their errors with his unusual dealing of multiplication procedures. They neverroduced procedural errors on the same material administered to PN. The fewrrors they committed were in fact and/or rule retrieval.

While in educated participants only sporadic errors commonly emerge inulti-digit multiplications, a huge variability in the quantity of errors is found

n multi-digit divisions. Division is, indeed, seldom performed manually afterchool, and most people, having lost familiarity with it, need to take time foretrieving and work out from memory the appropriate procedure. For this reason,
n the lack of recent positive evidence of intact performance, division is seldomnvestigated in neuropsychological work, and was excluded from this study.
Both the first and the second assessment were carried over several sessions.he type of operation and the order of presentation of each operation was bal-nced over sessions.

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as a consequence of the rupture of a right middle cerebral artery aneurysm.

. First assessment

.1. Neuropsychological background

PN spontaneous speech was fluent, with no aphasic signsn either production or comprehension (Aachen Aphasia Test,talian version, Luzzati, Willmes, & DeBleser, 1991). Repetitionnd automatic speech were normal. His oral comprehension ofords and sentences assessed was perfect. Both reading andriting were affected by visual neglect. He performed well onnaming task (score = 76; mean score = 64; S.D. = 2; BORB)

nd on a verbal phonological fluency task (score = 35.8; meancore = 36.28; S.D. = 10.81, Carlesimo, Caltagirone, Gainotti, &ocentini, 1995).His digit span forward was well preserved in the auditory

odality (span = 6) and his visuo-spatial span was within nor-
al limits, though at the lower level (Corsi span = 4, mean
core for young adults = 5.11; S.D. = 1.01; testing material andorms taken from Spinnler & Tognoni, 1987). The evaluationf his long-term memory showed learning and retrieval diffi-

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Table 1PN’s results (N of correct) obtained in the number processing and calculationassessment performed at 8 and 12 weeks after the onset of the disease

Subtest First period Second period

1. Dots counting 18/20 20/20

2. TranscodingRepetition 15/15 15/15Written verbal → spoken verbal 15/15 15/15Spoken → written verbal 15/15 15/15Arabic → written verbal 20/20 20/20Arabic → spoken verbal 20/20 20/20Written verbal → arabic 20/20 20/20Spoken verbal → arabic 20/20 20/20

3. Recognition of arithmetical signs 12/12 12/12

4. Number comparisonWritten verbal numerals 20/20 20/20Arabic numerals 20/20 20/20

6. Writing down an operation 8/8 8/8

7. Arithmetical factsMultiplication 41/44 44/44Addition 25/25 25/25Subtraction 23/25 25/25

8. Multiplication multiple choice 34/36 36/369. Written calculation (three addition,

three subtraction, threemultiplication)

2/9 4/9

10. Approximate calculationAddition 0/3 0/3Subtraction 0/3 0/3Multiplication 2/3 2/3Division 0/3 0/3

11. Mental calculationAddition 1/3 1/3Subtraction 2/3 2/3Multiplication 1/3 2/3Division 0/3 1/3

12. PrinciplesAdditions 4/15 14/15

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ulties. His performance on the Rey auditory-verbal learningest was within the norms for immediate recall (score = 34.3;

ean score = 50.12; S.D. = 8.35; Carlesimo et al., 1995) but pooror delayed recall (score = 4.2; mean score = 11.37; S.D. = 2.17;arlesimo et al., 1995). In the Babcock story recall PN was

mpaired (score = 9.34; cut-off = 15.76; De Renzi, Faglioni, &uggerini, 1977), also showing some difficulties in focusingttention on the material to be stored. In fact, his attentionalunctions were partly disturbed. Backward digit span was poorspan = 3) as well as selective visual attention (score = 16; meancore for young adults = 53.54; S.D. = 6.76; Spinnler & Tognoni,987).

An impaired performance was found in the Albert’s testscore = 21; Albert, 1973), in the Bell test (7 out of 35; Gauthier,ehaut, & Joanette, 1989) and also the letter cancellationevised by Diller and Weinberg (1977) was greatly affected bynilateral neglect (the difference between the omission on theight and the left side of the sheet was 37; cut-off ≥ 3, Vallar,usconi, Fontana, & Musicco, 1994). Moreover, when PN was

equired to draw from memory a clock he displaced all the hourso the right. Instead, no imaginal unilateral neglect was detectedhen PN was required to describe the mental image of some

amiliar place from a given vantage point (e.g., tests equiva-ent to Bisiach’s & Luzzatti, 1978, “Duomo di Milano” test).N also did not show any sign of personal neglect detectableith available techniques (Beschin & Robertson, 1997). A mild

onstructional apraxia was also observed, while no ideomotorr ideational apraxia was detected. Visual discrimination wasormal (nine out of nine on Poppelreuter’s test). Average scoresere found in telling differences between words and in interpre-

ation of proverbs and criticisms of absurd stories (score = 40.25;ean score for young adults = 51.93; S.D. = 6.22; Spinnler &ognoni, 1987).

The first neuropsychological assessment included the admin-stration of an experimental battery for number processing andalculation; the results are presented in Table 1.

PN showed a well preserved ability in all transcoding tasksetween the Arabic, alphabetic and phonological codes. Hisomprehension of both Arabic and written verbal numerals upo five-digits was flawless as indicated by his performance inhe number comparison task. He was able to recognise the fourrithmetical signs and to write down an operation. Althoughometimes slow, he demonstrated a preserved knowledge ofrally presented arithmetical facts including both fact- and rule-ased problems. The former are assumed to be directly retrievedrom long-term memory where they are stored in the form ofndividual representations (e.g., 5 + 2, 7 × 5, 8–3), while theatter are assumed to be stored in the form of general rulese.g., N × 1 = N; 0 × N = 0, for any number). A multiple-choiceask of multiplication facts where he had to choose the correctnswer among alternatives (measured through the calculationattery from Delazer, Girelli, Grana, & Domahs, 2003) waslso well performed. Multi-digit calculation for additions, sub-
ractions and multiplications was instead clearly impaired (22%orrect). The errors reflected a failure to use the carry and bor-ow procedure. Interestingly, spatial errors (e.g., misalignmentf the numbers in their appropriate columns) for multiplica-
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Multiplications 7/15 5/15

ions problems were also observed. Moreover, approximationmeasured through the calculation battery from Delazer, Girellit al., 2003) was impaired: PN was unable to choose betweenour alternatives the closest to the correct result to a multi-digitalculation task. Mental calculation (measured through the cal-ulation battery from Delazer, Girelli et al., 2003) was insteadood for subtractions but clearly impaired for additions, multi-lication and divisions. Finally, his conceptual knowledge (mea-ured through the calculation battery from Delazer, Girelli et al.,003) was severely compromised as evidenced by the arithmeticrinciples task (e.g., 24 + 37 = 61; 37 + 24 = ?; or 12 × 4 = 48;2 + 12 + 12 + 12 = ?).

In summary, the preliminary assessment of PN’s numeri-
al skills indicated well preserved number processing abilities,ogether with specific difficulties in calculation. The good per-ormance in arithmetical facts, in contrast with impaired multi-igit written calculation, was the starting-point for further testing

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and investigations, where multiplication problems and relatedspatial errors were studied more in detail.

4.2. Experimental tasks

4.2.1. Definition and description of the four elementaryarithmetical operations

A definition and, subsequently, a description of procedureswas required for all basic arithmetical operations.

4.2.2. Multi-digit operationsEighty-five multi-digit operations (23 additions, 31 subtrac-

tions and 31 multiplications) were visually presented to beanswered in written modality. Operations varied in complexityin terms of carrying or borrowing requirements. When “0” wasincluded, its relative position within the numerals was balancedacross operations (e.g., NNN × N0-N; NN0- × NNN, etc.). PNwas asked to work out the solution of each operation with notime pressure and was encouraged to comment on the opera-tional steps used.

Addition. PN was presented with 23 multi-digit additions; 16of them required the carry procedure. The following structureswere included: 2 + 2 digit (N = 2); 3 + 2-digit (N = 3); 3 + 3-digit(N = 9); 4 + 2-digit (N = 2); 4 + 3-digit (N = 7). Operations werepresented in random order.

Subtraction. PN was required to solve 31 multi-digit subtrac-tions; 20 of them required the borrowing procedure. The fol-lowing structures were included: 2–2-digit (N = 9); 3–2-digit(N = 13); 3–3-digit (N = 9). Operations were presented in ran-dom order.

Multiplication. PN was required to solve 31 multi-digit mul-tiplications; 22 of them required the carrying procedure. Thefollowing structures were included: 2 × 1-digit (N = 1); 2 × 2-digit (N = 4); 3 × 1-digit (8); 3 × 2-digit (N = 10); 3 × 3-digit(N = 6); 4 × 2-digit (N = 2). Operations were presented in ran-dom order.

4.3. Results

4.3.1. Definition and description of the four elementaryarithmetical operationsAddition. “Is the numerical sum of a number of digits, for exam-ple 6 + 6 equals 12 or 60 + 60 equals 120”. “One has to take anumber and add it; for example, if you were to explain a childhow much 2 + 2 makes, take two fingers and give him two otherfingers and then you tell him to count how many fingers are there.

Subtraction. “Is a mathematical calculation which foresees totake away numbers from other numbers”. “One needs it whenone wants to take away numbers from other number; for example12 − 12 makes 0, there is no rest, 12 − 1 makes 11, etc.”.

Multiplication. “One has to take numbers and multiply them by
other numbers”. “One needs to take some numbers and multiplythem by other numbers. It is quicker than an addition, becausethere is an exponential adding. For example 4 × 2 is 4 + 4 and4 × 4 is 4 + 4 + 4 + 4. It is a quicker calculation”.
gia 44 (2006) 2972–2986

ivision. “This is the inverse of a multiplication; one has to takeumbers and divide them by other numbers”. “If we divide, it ishe opposite of a multiplication; for example, 16 is divided byand makes 4 because if I take 4 and multiply it by 4 I get the

same] result”.

.3.2. Multi-digit operationsSometimes PN produced more than one error on a single

peration; thus the number of errors was larger than the num-er of operations wrongly produced. The results are presentedn Table 2. The three controls participants committed a totalf four arithmetical fact retrieval errors and one rule retrievalrror.ddition. PN’s performance was good: he correctly solved 19perations (82.6% correct) and thus produced only four errors:wo missed carry and two omissions of digits.ubtraction. PN solved correctly 18 operations (58% correct).e made 17 errors: 9 missed borrow, 4 arithmetic facts mistakes,counting errors and 1 digit omission.ultiplication. PN was severely impaired: all multiplication

perations were wrongly produced (0% correct). He made16 errors that were analysed and classified according tohe step in which they occurred (i.e., factor selection; com-utation of the partial products; computation of the finalesult) and to the sub-procedures neglected or inappropriatelypplied (e.g., carry, spatial alignment; retrieval of arithmeticacts).

The following types of errors were observed in the successiveteps:

a) Factor selection: The numbers to be multiplied always cor-responded to the correct pair: however PN produced 32errors (14.8%): 25% of them (N = 8) were omissions of adigit on the left side and 75% of them (N = 24) consisted ofselection and multiplication of a factor “0” not existing inthe actual multiplication. Thus, he performed an extra stepthat is not required by the multiplication algorithm: writingunnecessary zeros was in this case considered an error. Anerror of this sort does not lead, of course, to a wrong finalresult. If, as in the examples reported in Fig. 2a and b, thefinal result was wrong, such outcome was due to other typesof errors. Note that the patient shows a perfect knowledgeof zero-rules (N × 0 = 0, etc.). In three cases PN multipliedone digit of the multiplicator with an inexistent “0” at theleftmost side of the multiplicand (Fig. 2a) while in 12 caseshe multiplied an inexistent “0” at the leftmost side of themultiplicator with one digit of the multiplicand (Fig. 2a). Healways performed this extra step when all digits of the multi-plicand were already multiplied. He then wrote the obtainedresult, “0”, in the partial products. In nine cases, instead, PNselected an inexistent factor “0” on the leftmost side of themultiplicator and multiplied it with another inexistent fac-tor “0” on the leftmost side of multiplicand (Fig. 2b). Again
he wrote the obtained result “0” in the partial products. Healways verbalised all procedural steps. Note the PN alwaysobserves the prescription of proceeding from the right digitleftwards, a part of the procedure that multiplication has

A.G

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europsychologia44

(2006)2972–2986

2979

Table 2Number of correct responses and of errors in solving multi-digit written operations. PN’s distribution of error types in multiplication was divided according to the step in which the errors occurred and to thesub-procedures neglected or inappropriately applied (e.g., carry, spatial alignment; retrieval of arithmetic facts) taking into account the larger first and the smaller first condition in the first and in the second assessment

2980 A. Grana et al. / Neuropsychologia 44 (2006) 2972–2986

Fig. 2. Examples of PN’s errors in the factor selection (note that other errorsnot involving factor selection are also present): (a) multiplication of one digitin the multiplicator with a non-existent factor zero at the leftmost side of themultiplicand (e.g. he said: “0 × 4 is 0, write 0; 0 × 6 is 0, write 0; 0 × nothing is0, write 0”) and multiplication of a non-existent factor zero at the leftmost sideof multiplicator with one digit in the multiplicand (e.g., he said: “nothing × 4, is0, write 0; nothing × 6, is 0, write 0; nothing × 4 (that is the carry of 8 × 6”) is 0,write 0; nothing × 3 (that is the carry of 8 × 4), is 0, write 0”); (b) multiplicationof a non-existent factor zero in the multiplicator with a non-existent factor zeroin the multiplicand (e.g., he said “6 × 0, is 0; 6 × 0 is 0; 6 × 5 is 30, write 0 andcarry 3 (that is write near the 0), nothing x nothing is 0, write 0”); (c) selectionand multiplication of a factor zero that does not exist in the actual multiplicationand is not written in the partial product (e.g., he said: “2 × 6 is 12, write 2and carry 1; 2 × 0 is 0 + 1 (carry) is 1”); (d) multiplication of one digit in themultiplicator with a non-existent factor zero (e.g., he said: “3 × 3 is 9, write 9;3 × 1 is 3, write 3; 3 × nothing, is 0, write 0; 1 × 3 is 3, write 3; 1 × 1 is 1, write1; 1 × nothing, is 0, write 0; 1 × 3 is 3, write 3; 1 × 1 is 1, write 1; 1 × nothing,i

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Fig. 3. Examples of PN’ errors produced in the calculation of partial products(note that other errors not involving calculation of partial products are alsopresent): (a) carry reported on the leftmost side of the multiplicand; (b) carrywritten next to the partial product; (c) inappropriate decomposition of the lastretrieved arithmetical fact in the row and carry written near to the partial product(e.g., he said: “6 × 4 is 24, write 4 and carry 2”); or (d) added to the next partialproduct in the row just below (e.g., he said: “. . .4 × 5 is 20, write 0 and carry2; 6 × 2 is 12 + 2 (carry) is 14, write 4 and carry 1; 6 × 5 is 30 + 1 (carry) is 31,write 1 and carry 3; 7 × 2 is 14 + 3 is 17, write 7 and carry 1. . .”); (e) omissionsoa

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5

5.1. Neuropsychological background

s 0, write 0”).

in common with both addition and subtraction (which he,indeed, was eventually, able to correctly perform).

b) Calculation of partial products: Once factors to be multi-plied were correctly selected, errors occurred in the calcula-tion of partial products (114/216, 52.8% of the total errors).Seventy-eight errors (68.4%) could be attributed to inap-propriate use of carry. PN always wrote down the carry (astrategy children are instructed to use in school) but neveradded it to the next partial product. In 28 cases he reportedthe carry on the leftmost side of the multiplicand (Fig. 3a)while in 37 cases he wrote it next to partial product (which ishow the instruction goes; Fig. 3b). Finally, in 16.7% of cases(N = 13) he decomposed inappropriately the last retrievedarithmetic fact in the row (i.e., the only one that does not
need decomposition) and wrote the carry near to the par-tial product (Fig. 3c). Other errors consisted of omissions ofthe auxiliary line (always when it was required) resulting in t
f the auxiliary line. Note that this error type is present also in the examples bnd d.

failure in the spatial alignment of the spatial products (22%;N = 25) or in failure of arithmetical fact retrieval (2.6%;N = 3). Few errors (7% error; N = 8) were hard to interpretand were classified as “other”.

c) Sum of partial products: PN produced 70/216 errors(32.4%). Such errors resulted from spatial misalignment.In this case the error was determined by the inclusion of thecarry written near to partial product (N = 11; see Fig. 4a)or by further addition of carries mixed to digits in partialproducts (N = 59; see Fig. 4b).

. Second assessment

Critical tests were repeated in this second assessment, withhe addition of other tests aimed at deepening the analysis of the

A. Grana et al. / Neuropsycholo

Fig. 4. Examples of PN’s errors in the sum of partial products (note that othereod

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atient’s deficit. PN was thus retested on all previously adminis-rated tests on visual neglect, that were now performed without

istakes; in addition he was administrated the behavioural inat-ention test (Wilson, Cockburn, & Halligan, 1987). Scores foronventional sub-tests were within the norms (score = 135): nolinical signs of visual neglect were thus detected with thisdditional test. A more detailed analysis on object and spaceerception was also carried out though the visual object andpace perception battery (Warrington & James, 1991). Thisssessment showed a weak performance only in the numberocation and cube analysis subtests. Raven coloured progressive

atrices were administered, obtaining a normal overall scorescore = 27; mean score = 30.03; S.D. = 4.11; Carlesimo et al.,995).

The general assessment for number processing and calcu-ation battery was also repeated (see Table 1). Overall, PN’serformance improved in all tasks with the exception of approxi-ate calculation, mental addition and multiplication conceptual

nowledge. Multi-digit written calculation was still impaired.

.2. Experimental tasks

.2.1. Multi-digit operationsAt this time PN was less impaired, but, while he was perfectly

ble to perform multi-digit additions and subtractions, he keptommitting a considerable amount of errors on multiplications.

gia 44 (2006) 2972–2986 2981

One hundred and fourteen multi-digit operations (6 additions,subtractions and 102 multiplications) were visually presented

o be answered in writing.ddition. Six multi-digit operations with the structures 3 + 2-igit (N = 2), 3 + 3-digit (N = 2) and 4 + 3-digit (N = 2) were visu-lly presented in random order. All of them required carrying.ubtraction. Six multi-digit operations with the following struc-ure 2–2-digit (N = 2); 3-2-digit (N = 2); 3–3-digit (N = 2) wereisually presented in random order. All of them required bor-owing.

ultiplication. PN was visually presented in random order with02 multi-digit operations varying in complexity in terms ofarrying requirements. The order of operands was also manipu-ated: 49 multiplications were presented with the larger num-er in the multiplicand (e.g., NNN × N) and 53 were pre-ented with larger number in the multiplicator (e.g., N × NNN).hus, among 102 operations, the following structures were

ncluded: 2 × 1-digit (N = 9); 3 × 1-digit (N = 9); 3 × 2-digitN = 17); 4 × 1-digit (N = 12); 4 × 2-digit (N = 2) and 1 × 2-igit (N = 9); 1 × 3-digit (N = 9); 1 × 3-digit (N = 9); 1 × 4-digitN = 12); 3 × 2-digit (N = 17); 4 × 2-digit (N = 6).

He was again required to write down each operation with noime pressure and was encouraged to comment on the operationalteps used.

.3. Results

.3.1. Multi-digit operationsAt this time PN was less impaired, but, while he was perfectly

ble to perform multi-digit additions and subtractions, he keptommitting a considerable amount of errors on multiplications.he results are presented in Table 2. Control participants com-itted eight fact retrieval errors, three on multiplications with

arger number in the multiplicand and five on multiplicationsith larger number in the multiplicator.ddition. PN’s performance was error free.ubtraction. PN scored fast and flawless.ultiplication. Again he produced more than one error on a

ingle operation: thus the number of errors was larger than theumber of operations wrongly produced. Overall, PN was cor-ect in only 58.8% of the cases. He mistook 15 multiplications inhe “larger first” arrangement (69.4% correct) and 27 multiplica-ions in “smaller first” arrangement (49% correct). He produced

total of 197 errors: 40 of them in “larger first” and 157 onsmaller first” operations.

Errors were separately counted for the two kinds of arrange-ent and analysed according to the classification used for therst set of tasks; thus the following types were observed:

“Larger first” multiplication:(a) Factor selection: PN made nine errors resulting from

omissions of the leftmost digits (N = 3) or selection andmultiplication of a factor “0” not existing in the actual
multiplication (N = 6). This last extra step, not requiredby the multiplication algorithm, was performed multiply-ing one digit of the multiplicator with an inexistent factor“0” of the multiplicand but without writing the result. The

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obtained result performed when PN decomposed the lastretrieved arithmetical fact (when not appropriate) was notwritten but verbalised and occurred only once all digitsof the multiplicand were already multiplied (Fig. 2c).

(b) Calculation of partial products: PN produced 30 errorsin this step. 12/30 errors could result from inappropriateuse of carry. While in one case he put down the carry nearto the partial product without adding it to the next par-tial product, in 11 cases he decomposed inappropriatelythe last retrieved arithmetical fact of each row: in 4 ofthese cases he added the carry to the next partial productin the row just below, while in 7 cases he put down thecarry near to the partial product. Other errors consistedin omissions of the auxiliary line resulting in failure inthe spatial alignment of the spatial products (N = 4); infailure of arithmetical fact retrieval (N = 3) or in count-ing errors (N = 5) when he added the carry to a retrievedarithmetical fact. Six errors were difficult to interpret andwere classified as “other”.

(c) Sum of partial products: PN produced only one errorresulting from spatial misalignment involving a carrywritten near to the partial product and the same partialproduct.

“Smaller first” multiplication:(a) Factor selection: PN produced 49/157 errors: two were

omissions of a digit on the left side while 47 resulted,as in the “larger first” arrangement, from selection andmultiplication of factor “0” not existing in the actual mul-tiplication. As argued above, this last extra step is notrequired by the multiplication algorithm and was per-formed multiplying one digit of the multiplicator withan inexistent factor “0” of the multiplicand. In 37 casesthe obtained result “0” was written in the partial products(Fig. 2d), while in 10 cases it was not written but ver-balised. This extra step occurred only once all digits ofthe multiplicand were already multiplied.

(b) Calculation of partial products: PN produced 97/157errors. The errors could result from inappropriate use ofcarry (N = 68) when he decomposed inappropriately thelast retrieved arithmetic fact of each row. Particularly, innine cases he added the carry to the next partial productin the row below (Fig. 3d); while in 59 cases he wrote thecarry near to the partial product. Other errors consisted ofomissions of the auxiliary line, resulting in failure in thespatial alignment of the spatial products (N = 22; Fig. 3e);in failure of arithmetical fact retrieval (N = 2) or in count-ing errors (N = 2) when he added the carry to a retrievedarithmetical fact. Three errors were difficult to interpretand were classified as “other”.

(c) Sum of partial products: PN made 11/157 errors consist-ing in spatial misalignment between a carry written near tothe partial product and the same partial product (Fig. 4c).

In summary his main difficulties in written multiplicationcarry procedure, alignment of intermediate products, factorelection) have been amplified when operations where presented

gia 44 (2006) 2972–2986

n the “smaller-first” arrangement compared to his performancen the “larger-first” arrangement.

. Discussion

The present paper describes a patient, PN, with an exclu-ively right sided lesion, who showed, upon administration ofseries of mathematical tasks, an interesting, previously unre-orted, pattern of neuropsychological dissociations in perform-ng multi-digit written multiplication. In contrast with availableeports, PN’s study was conducted, longitudinally, on a sizeableample of operations which allowed to observe very consis-ent phenomena. PN’s pattern of performance is worth reportingndependently of possible interpretations, that per se may be pre-

ature: however it will be only through descriptions of this sortnd careful comparison among cases that advancement could bexpected.

PN showed preservation of arithmetical facts and rule-basedrithmetical problems as well as of procedures in addition as wells subtraction (these last only in the second administration): inontrast, his multiplication procedures were severely affected.his notwithstanding, PN was able to describe the procedures inords, and seemed to know the various steps that are necessary

o completely perform the operation. However, he committed aumber of interesting errors.

Through a careful error analysis, conducted for the first timen a step by step fashion, it is indeed possible to isolate an impor-ant portion of PN’s errors that seems to be of a specificallypatial nature, inherent to the demands of a multi-digit multi-lication. These errors can be distinguished from other typesf errors, including those, expected after a right hemisphereesion, determined by a generic inability to deal with spatial

aterial, or from other deficits, like neglect, affecting cogni-ive capacities across the board. Indeed it may not be possibleo eventually demonstrate that such errors exclusively affecthe calculation domain. It is important, however, in absencef a clear proof of the contrary, and vis-a-vis a theory thatxplains them in calculation-specific spatial terms, to establishhat a generic spatial deficit is unlikely to be involved in theirrigin.

PN’s previously unreported type of errors, not committedy matched controls, may thus be summarised as follows, alongith the reasons why they may be considered as spatial in naturer, at least, resulting from the application of strategies aimed atompensating for a spatial difficulty:

a) Selection and multiplication of a factor “0” that did not existin the actual multiplication, then adding the carry to the left-most side of the multiplicand or of the partial products. Thistype of error is indeed puzzling: however it may be con-sidered as a compensation to a spatial difficulty, insofar the“0” occupied a position at the leftmost border of the fac-tors, when presumably the patient did not know where to
proceed. Adding “0” may therefore have been a strategy tokeep a border and stop. This may not be the ultimate originof this strange error. But, indeed, a memorised, well spec-ified spatial layout, clearly deficient in the patient, would

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b) Adding the carry to the leftmost side of the multiplicand orof the partial product. Thus, while the patient knew whatthe carry was (and if there was any), he seemed to haveforgotten the fact that there is not an explicit place for itin the canonical spatial layout of the operation. As com-pensation he systematically wrote down the carry withoutadding it to the next partial product. This error could per-haps be explained simply by positing that the patient couldnot remember what is supposed to be done with the carrydigit. The precise origin of such error is hard to understand:however, a full knowledge of the proper spatial layout of theoperation would have certainly prevented in this case thisparticular compensatory strategy and therefore this form oferror.

c) Inappropriate decomposition of the last retrieved arithmeticfact in the row, and carry written near to the leftmost sideof the partial product or added to the partial product in therow below. He knew when and how the carry must be addedbut he seemed to have forgotten where. As compensationhe added it nearby, besides or below. The patient seems toignore that the spatial layout requires independent calcula-tion in each row.

Most of these explanations may, in principle, be objected to:he procedures used by PN may, in fact, be rejected on purelyogical grounds. An exhaustive and logically determined con-rol of all necessary steps is however unlikely to be working in

ulti-digit calculation unless specifically decided. In expert butot theoretically minded calculators, highly automatised pro-esses may instead dispense with such a strict control of logicalteps. Even some expert calculators, in fact, would be at loss inpelling out verbally the logic behind the spatial display of theumbers they can quickly, mindlessly but efficiently, lie downn paper in a complex operation. It is therefore simpler to pos-ulate that, normally, human calculators are rather relying on aell established routine. Such routine may be based on an over-

earned spatial layout or spatially organised specific procedure.precise, automatically running, memory for this layout is prob-

bly missing in PN, who seemingly resorts to a substitute wrongorm of it or to failing strategies to keep the operation going. Inther words it is important to stress that, while attentional andorking memory disorders may have deprived PN of important

ources to monitor his multiplication, he seems to lack the pos-ibility of at least partially compensating them via resorting to aess demanding spatial procedural knowledge (normals, indeed,t should not be forgotten, may need spatial procedural knowl-dge when working memory demands, in complex operations,xceed their own capacity).

Even if the above reported hypotheses were considered toopeculative or definitively wrong, there are further proofs of PN’specific spatial problems. Convincing evidence of the spatial
ature of all PN’s errors in multiplication comes, indeed, fromhe fact that his performance clearly worsened in the “smaller”rst condition. “Smaller first” and “larger first” conditions doot differ from each other in the number of logical steps. The
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gia 44 (2006) 2972–2986 2983

smaller first” condition, is more demanding in terms of thepatial arrangement. In the learning stage is certainly taughtnd consequently acquired later than the “larger first” conditionHofer, Grana, & Semenza, 2003). Indeed, these very reasonsake the “smaller first” arrangement unfamiliar. Unless required

o do it in school, most people would presumably set the opera-ion with the larger number on top. But this would be precisely tovoid complications determined by a more complicated spatialrray. Everything else, including the reasoning requirements,s exactly the same. Consider, for instance, the extreme case,hown in Fig. 3, of 2 × 134: after multiplying 2 × 4 and writ-ng 8, one must shift one column to the left before writing the; however there is nothing in this column on the precedingine—one must write the 6 underneath a blank region of therevious line, which may cause uncertainty about where indeedo write the digit. Hofer et al. (2003) independently observedearners’ strategies with respect to this difficulty: most learners,ndeed, successfully overcome it just by strictly adhering to thepatial rule of column shifting, rather than by being explicitlyuided by logical reasoning and decide that at that point theecade column should be filled in. It should be remembered thatN always verbalised what he was doing, which permitted thexaminer full awareness over his strategy.

In this respect, mind that the evidence that the patient pre-erved the “what” does not rest only on his good definition of aultiplication but on his telling explicitly (upon explicit instruc-

ions), in a step-by-step fashion, what had to be done (i.e., henew what carrying is and in fact could do it in addition!). It isertainly possible to work out the right procedure on a reasoningasis only, but it would be painstaking and probably not withinverybody’s ability. It is contended here that normal calculatorsas well as PN) do not do it this way: what they do is to apply theperation of carrying where an overlearned procedure dictates.n any case carrying were not the only errors.

Besides the above listed errors, PN also made errors that haveeen already described in right hemisphere spatial acalculia: (a)mission of a digit on the left side; (b) omission of the auxiliaryine; (c) column confusion. These errors can be easily isolatednd distinguished from the ones listed above, and can certainlye considered as due to failure in the spatial layout specific forultiplication, once generic across-the-board spatial problems

re ruled out.Errors due to across-the-board spatial difficulties cannot

ndeed be claimed to be at the basis of all PN’s problems. While,n fact, it is not possible to exclude the influence on PN’s per-ormance of spatial disorders of the type known to affect rightemisphere patients, it is possible to contend that such genericisorders of to space or of space representation are not sufficiento produce all the observed types of errors.

In fact, a generalised disorder in the attention to space wouldardly produce systematic errors, but would instead leave theatient helpless in his space exploration: as a consequence theisplay of the operation would reflect more or less chaotic
ttempts equally affecting every portion of the operation, andot just specific steps (why, indeed, should the spatial arrange-ent of some steps be systematically more resistant to attention
roblems?). PN was not particularly prone to attention problems.

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Equally, PN’s deficit with multiplication procedures couldot possibly stem from the more restricted attention disorderescribed under the heading of “monitoring” deficit (Semenzat al., 1997): in fact PN was consistent in his errors, always endedhe operation, and his mistakes happened only at certain levelsnd did not increase with the progress of the operation.

Unilateral spatial neglect (a likely symptom in a right hemi-phere lesion, but clinically absent in every form when the sec-nd series of tasks was administered) would also have affectedomplex, multi-digit addition and subtraction (carrying is cer-ainly not more complex than borrowing), but these operationsere not disturbed at least at the time of the second evaluation. Aossible objection may be that multiplication is more demandingnd likely to uncover neglect in a milder form. However, in mul-iplication PN makes the same kind of mistake even in simplerperations like (234 × 6). Why, in this case, carrying in additioniffers from carrying in multiplication? Only because carryingn multiplication happens in a different context: according tohe learned spatial procedure one has to operate in a differentpace display: there is no mathematical or conceptual differencehatsoever and, as just said, there are cases where multiplica-

ion cannot be considered more spatially demanding; PN’s mainroblem is rather that he forgot where to apply the operation step.

There is, however, further proof against a significant contri-ution of neglect to the whole of PN’s spatial symptomatology.n fact, a distinctive type of error committed by PN consistedf explicitly writing the carry to the leftmost side of the multi-licand as well as of the partial products. The presence of a leftided neglect would dictate writing carry notes to the right, or,owever, not on the left. This behaviour cannot derive by an auto-atic procedure because it is not something that is learned. To

e true (see above) PN also made a few omissions of digits in theeft side, but their number was negligible and this type of erroras virtually absent at the time of the second administration.Another hypothesis to keep into account is that PN’s errors

ay have originated from a disturbance of a spatial workingemory component, a sort of “calculation sketching pad”. While

t may be impossible to entirely exclude the influence of a limita-ion in such putative mechanism, that may certainly contribute toN’s difficulties, this is still an unlikely explanation. In fact PNas, though at the lower level, within norms at Corsi’s test, tes-

ifying a sufficiently efficient spatial memory span; furthermoreis errors were not casually dispersed.

In summary, the best explanation for PN’s problems cannot beound in a generic deficit of spatial processing or of spatial atten-ion. Such deficits may not be entirely ruled out for PN. Indeed,here would never be enough testing for a sceptical neuropsy-hologist to exclude these generic sources deficits in any givenatient. However, negative symptoms should not be the onlynes guiding the interpretation of neuropsychological patients’erformance. More information is carried by positive symptoms.N’s difficulty with multiplications may indeed reflect his lackf generic resources but his non-casual errors point the fact that
e cannot (unlike normals that have been thought to do so) makep for his deficit with the spatial framework or spatial layout thatelps in conducting complex operations correctly to an end. Inther words, if PN had an intact knowledge of the spatial layout,
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e would be able to compensate for the calculation problemse displays, irrespective of their origin. The observation of suchase makes it compulsory analysing future cases at this levelf detail. It will be important trying to answer questions as theollowing: to what extent to know “where” certain operationteps should be displayed in writing is based on spatial cog-ition? How can one distinguish between knowledge of thesespatial facts” or “spatial procedures” and visuo-spatial repre-entations? How optional is to resort to spatial knowledge inarrying out a complex arithmetical operation? Most of the anal-ses made in this study have never been applied before and thushe evidence collected in this particular case may not be entirelyompelling. It is nevertheless intriguing to suggest that patientsike PN might have difficulties in relying on a stored layout rep-esentation specific to multiplication. This representation woulde normally activated in a (quasi-)automatic fashion in ordero support spatially arranged complex calculation and dispenseith more demanding controlled processing based on logical cri-

eria. As already mentioned, a representation of this sort has beenostulated by Dehaene and Cohen (1995) along with a prelimi-ary description. PN’s difficulties in the carry procedure matchell with this assumption, because PN knows that a retrieved

act has to be decomposed but fails to add it to the next fact in theight place. Thus, PN seems to fail in intermediate and “borderteps” in a multiplication, while there is evidence that he knowshat steps to make. All these errors may be typified as resulting

rom the attempt to compensate to the lack of a guiding spatialcheme.

A speculative but likely conclusion is therefore that, vis-a-is his pattern of performance in multiplication procedures, PNoes know what, when and how to carry out the various steps,ut he does not know where. What he may lack is a spatialchema of multiplication. This putative spatial schema can beeen as the sum of spatial markers, which might be necessary toost calculating subjects, especially in intermediate steps and or

border steps”. Such schema might thus help in keeping the sin-le sub-steps together by representing the information of wherexactly each sub-step should be placed. Next case studies onatients similar to PN should be aimed at testing these hypothe-es. Equally, PN’s case points out to the necessity of collectingurther evidence in the learning domain: learners’ strategies toope with logical and spatial aspects of complex multiplica-ion must be further disentangled. As some authors active inhe first half of last century (Ehrenwald, 1931; Peritz, 1918)ad suspected, on the basis of observations in the clinical andevelopmental domain as well as on phenomenological consid-rations (inspired by works such as those of Bergson (1911) orertheimer (1912)), there might, after all, be a primary spatial

calculia. Whether affecting functions that are normally storedn the right hemisphere, as PN’s case seems to suggest, is stilln open question.

cknowledgements

We wish to thank Prof. Paolo Di Benedetto and Dr. Emanueleiasutti of the Istituto di Medicina Fisica e Riabilitazione “Ger-asutta”, Azienda per i Servizi Sanitari, no. 4 “Medio Friuli”,

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taly for their support throughout this study. The present researchas been supported by the European Union (Marie Curie Actionontract “NUMBRA” 504927) and by a MIUR grant 2005 toarlo Semenza.

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